Path integral solutions for n-dimensional stochastic differential equations under α-stable Lévy excitation

In this paper, the path integral solutions for a general n-dimensional stochastic differential equations (SDEs) with $\alpha$-stable Lévy noise are derived and verified. Firstly, the governing equations for the solutions of n-dimensional SDEs under the excitation of $\alpha$-stable Lévy noise are obtained through the characteristic function of stochastic processes. Then, the short-time transition probability density function of the path integral solution is derived based on the Chapman-Kolmogorov-Smoluchowski (CKS) equation and the characteristic function, and its correctness is demonstrated by proving that it satisfies the governing equation of the solution of the SDE, which is also called the Fokker-Planck-Kolmogorov equation. Besides, illustrative examples are numerically considered for highlighting the feasibility of the proposed path integral method, and the pertinent Monte Carlo solution is also calculated to show its correctness and effectiveness.